# Brocard circle

In geometry, the Brocard circle (or seven-point circle) for a triangle is a circle defined from a given triangle. It passes through the circumcenter and symmedian of the triangle, and is centered at the midpoint of the line segment joining them (so that this segment is a diameter).

## Equation

In terms of the side lengths ${\displaystyle a}$, ${\displaystyle b}$, and ${\displaystyle c}$ of the given triangle, and the areal coordinates ${\displaystyle (x,y,z)}$ for points inside the triangle (where the ${\displaystyle x}$-coordinate of a point is the area of the triangle made by that point with the side of length ${\displaystyle a}$, etc), the Brocard circle consists of the points satisfying the equation[1]

${\displaystyle a^{2}yz+b^{2}zx+c^{2}xy={\frac {a^{2}b^{2}c^{2}(x+y+z)}{a^{2}+b^{2}+c^{2}}}\left({\frac {x}{a^{2}}}+{\frac {y}{b^{2}}}+{\frac {z}{c^{2}}}\right).}$

## Related points

The two Brocard points lie on this circle, as do the vertices of the Brocard triangle.[2] These five points, together with the other two points on the circle (the circumcenter and symmedian), justify the name "seven-point circle".

The Brocard circle is concentric with the first Lemoine circle.[3]

## Special cases

If the triangle is equilateral, the circumcenter and symmedian coincide and therefore the Brocard circle reduces to a single point.[4]

## History

The Brocard circle is named for Henri Brocard,[5] who presented a paper on it to the French Association for the Advancement of Science in Algiers in 1881.[6]

## References

1. ^ Moses, Peter J. C. (2005), "Circles and triangle centers associated with the Lucas circles" (PDF), Forum Geometricorum, 5: 97–106, MR 2195737
2. ^ Cajori, Florian (1917), A history of elementary mathematics: with hints on methods of teaching, The Macmillan company, p. 261.
3. ^ Honsberger, Ross (1995), Episodes in Nineteenth and Twentieth Century Euclidean Geometry, New Mathematical Library, 37, Cambridge University Press, p. 110, ISBN 9780883856390.
4. ^ Smart, James R. (1997), Modern Geometries (5th ed.), Brooks/Cole, p. 184, ISBN 0-534-35188-3
5. ^ Guggenbuhl, Laura (1953), "Henri Brocard and the geometry of the triangle", The Mathematical Gazette, 37 (322): 241–243, JSTOR 3610034.
6. ^